First-Principles Calculation of Vibrational Raman Spectra in Large Systems: Signature of Small Rings in Crystalline ${\mathrm{S}\mathrm{i}\mathrm{O}}_2$

We present an approach for the efficient calculation of vibrational Raman intensities in periodic systems within density functional theory. The Raman intensities are computed from the second order derivative of the electronic density matrix with respect to a uniform electric field. In contrast to previous approaches, the computational effort required by our method for the evaluation of the intensities is negligible compared to that required for the calculation of vibrational frequencies. As a first application, we study the signature of 3- and 4-membered rings in the Raman spectra of several polymorphs of ${\mathrm{S}\mathrm{i}\mathrm{O}}_2$, including a zeolite (H-ZSM-18) having 102 atoms per unit cell.

Figure 1

Vibrational Raman spectra of various ${\mathrm{S}\mathrm{i}\mathrm{O}}_2$ polymorph powders. Measurements are from Refs. 22. Theoretical frequencies are rescaled by $+5\%$, and the spectra are convoluted with a uniform Gaussian broadening having $4.0{\mathrm{c}\mathrm{m}}^{-1}$ width.

Figure 2

Raman intensities projected on the breathing modes of various rings labeled 3-MRs and $4\mathrm{\text{−}}{\mathrm{M}\mathrm{R}\mathrm{s}}_x$ (see the text). For clarity, the overlap intensity ($I_{\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{p}}^{\nu }$ in the text) is shifted vertically.